Intro to 4-Vectors and the Spacetime Interval #SoME4

Transcript

How often do we hear the words complex and spacetime in the same sentence? Not often enough, but we should because the payoff, it's connected to electricity. This video is about complex numbers, space, and time. So, let me ask you something. What is a complex vector field? What about a complex four vector field? It's already hard enough to visualize higherdimensional fields, let alone complex ones. But here's the twist.

Mathematically speaking, the treatment is remarkably simple. You don't need to know much more than some basic concepts about geometry and a little complex analysis, like knowing what a complex exponential is. You know, e raised to the power of some imaginary quantity. That's it. With this one simple tool and by giving the exponent a specific structure, we'll see how much easier it becomes to grasp some of the most difficult concepts, not just in special relativity, but also in quantum mechanics.

And it all starts with something you might already recognize, the Lorent transformations. Here we're going to look at them through the lens of something called rotors. These are mathematical objects that are expressed not coincidentally using exponentials. But first, just as one of my favorite channels wisely points out in one of their videos, don't worry about the technical terms you don't understand yet. We'll get to each one of them.

As one of the great mathematicians, John vonoman said, "In mathematics, you don't understand things. You just get used to them." >> For now, all you need to know is rotors encode rotations, as the name suggests. We also call them transformations which is just a broader term but in this context they're pretty much interchangeable. So when we want to transform something like a geometric object say a rocket symbolically speaking we multiply it by a rotor on one side and by its multiplicative inverse on the other. It's as if each multiplication performs half of the rotation.

Now Laurent transformations typically involve two kinds of rotors. One type reflects velocity like these two and another type reflects orientation like these other two. But in this series, we're going to consider a third exponential, a simpler one, just a complex number. The idea is to use it to build a more complete rotor, one that captures not just the motion of massive particles, but also charged. Have you ever heard of U1 symmetry? How about SU2 or SU3? If you haven't, that's okay.

We're about to explore the math behind them and in the process we'll see how to simplify this structure by expressing it as the product of two four vectors. What happens when we apply this rotor and square it which surprisingly we'll get a real number despite all this mess? A real number we'll call an invariant quantity like mass times the speed of light in relativistic momentum. But guess what? It gets better. Let's say we define W as a wave vector with a frequency in wavelengths all divided by H bar and X as a four position vector. What do you think is going to happen if we take not just any derivative but the space-time derivative of this expression? Spoiler, we'll get a factor of I over H bar coming from here and here.

We can then move that factor to the left hand side and this is what's going to take us to quantum mechanics. This video is part of the summer of math exposition. Yes, we're the kind of people who do math, even in the summer. So, we're keeping it short. This is a beautiful and extensive topic.

I'm breaking it down into multiple chapters. This first video will cover everything up to this point. This was the introduction and for part two, I'll assume that you're a little bit familiar with this particular vector notation and the things we usually represent with three-dimensional vectors, things like positions, velocities, or accelerations. You'll also need to know how vector addition and subtraction work, what a dotproduct is and that it relates to projections, and what a crossroduct is and how it's often tied to rotations. Now, if you've never heard of geometric algebra before, you're in for something great.

It's one of the most powerful and underrated branches of math. At its core is something called the geometric product, and it's the key to almost all the simplifications we'll see. It's super easy. Take the three components of the first vector and multiply each by the second vector like standard multiplication. Then distribute those products.

Again, just like regular algebra, except this time, remember, order matters. In other words, commutation matters. It's not the same to have y * z as z * y, but they anti-commute, meaning you just flip the sign when you swap the order of two different basis vectors. And when a basis vector multiplies itself, the result is one. Now we'll swap these three terms, change the signs accordingly, convert these other three into ones, then reorganize once and again extract a common factor.

And now we rewrite any product of two different basis vectors as I * the 3. I promise it's going to make sense in just a minute. So what does this get us? It lets us rewrite the entire expression as the dotproduct between the two original vectors plus I * their crossroduct or even more compactly using the wedge product. And that leaves us staring at one of the most iconic expressions in all of geometric algebra. All these rules might seem arbitrary at first, but they start to make perfect sense when we think of the basis vectors as matrices.

This particular set of matrices is called the pi matrices and they satisfy every single rule. Pick any two of them and perform standard matrix multiplication. In this case, only these terms are non zero. we can factor out an I, leaving behind the same matrix representation as Z. And if we swap the two matrices, we get the same result but with a negative sign.

So those two rules, they're really the same and the others just analogous. Now pick any one basis vector and multiply it by itself. You'll get the identity matrix. Any basis vector. So these three actually behave like identity elements when squared.

Now let's take it a step further. Instead of using three 2x2 matrices to build vectors, we'll now use four 4x4 matrices which lets us construct the main building blocks of relativity. Four vectors. These are known as the direct matrices. Now the rules change slightly.

Gamma 0 * itself equals 1. That's the identity matrix. Gamma 1 2 and 3 when squared are equal to minus1. Any two different gamas anti-commute. Just like before, these are not all the rules, but we're good to go for now.

The simplest four vector we can build is the four position vector. It's just like a three-dimensional position vector, but with a twist. One component now represents time. whatever your clock is reading at a given moment multiplied by the speed of light. So, we still get units of distance.

The other three components are standard spatial coordinates measured with something like a ruler. That's it. A four-position vector on its own isn't very useful. But the difference between two four positions, that's where things get interesting. Subtracting four vectors is straightforward, just component-wise subtraction.

We'll use the delta symbol to represent these differences. Now, we're going to perform one of the simplest and most powerful operations in relativity, which is just squaring a vector. Similar to what we did with three-dimensional vectors, we just multiply each term by the full vector four times, then distribute these products. Again, taking into account that commutation matters. Swap these two gamas, flip the sign, and these two terms cancel out.

In fact, we could do the same for all the off diagonal terms. We're left with three components that square to -1 and one that squares to positive 1. We simplify what remains. And if you've seen special relativity before, you'll recognize this result. It's called the space-time interval, a plain old number, a real number.

It comes from squaring a four vector, and it's what we call an invariant quantity. Let's see what that means in action. Example time. Imagine a body with mass ming at a constant velocity from point A to point B. You and your team are standing still with your clocks and rulers.

When the body's at point A, you take the first measurement. Here's how far away point A is from you. And here's what your clock reads at that precise moment. When it reaches point B, you take the second measurement and subtract the two. Now you have a spatial distance between A and B and the time it took for the body to travel.

Keep in mind these numbers are just examples. Now picture a second T moving at constant velocity relative to you. They also record when the body is at A and B. And again measure position and time. But here's the catch.

Their measurements differ from yours. This part is what feels counterintuitive about relativity. Einstein showed us that space and time measurements are not universal. Two observers can and often will measure different distances and different durations for the same event. But here's the genius part.

There's something they'll both agree on. the square of those differences. This scalar quantity, the squared interval, is the same for everyone, no matter how they're moving. That's why we call it an invariant. This is special relativity in a nutshell.

Observers may disagree on position and time differences. They may even disagree on the velocity of the object, but the squared values, those are the same for everyone. To talk about velocity, though, we'll need to talk about derivatives, and that's the topic of our next video. All right. So far, we've introduced the basics of geometric algebra, explored how vector products work in this framework, and extended them to four vectors, the language of special relativity.

We've seen how space and time combined into a unified structure, and how that leads to invariant quantities like the space-time interval. But we're just getting started. In the next video, we'll take the next big step, introducing derivatives in spaceime. That's where things really start to move, literally. We'll see how four vectors evolve, how we define velocity in this framework, and how momentum and energy emerge from the math naturally, not just as real quantities, but complex.

Stay tuned. See you next time. The real there's such a lot in the world. There's so much distance between the fundamental rules and the final phenomena that it's almost unbelievable that the final variety of phenomena can come from such a steady operation of such simple rules. [Music]